pdf of paper Journal of Geoscience Education

To say Earth systems are complex, is not the same as saying they are a complex system. A complex system, in the technical sense, is a group of “agents” (individual interacting units, like birds in a flock, sand grains in a ripple, or individual units of friction along a fault zone), existing far from equilibrium, interacting through positive and negative feedbacks, forming interdependent, dynamic, evolutionary networks, that possess universality properties common to all complex systems (bifurcations, sensitive dependence, fractal organization, and avalanche behaviour that follows power-law distributions.)

Chaos/complex systems theory behaviors are explicit, with their own assumptions, approaches, cognitive tools, and models that must be taught as deliberately and systematically as the equilibrium principles normally taught to students. We present a learning progression of concept building from chaos theory, through a variety of complex systems, and ending with how such systems result in increases in complexity, diversity, order, and/or interconnectedness with time—that is, evolve. Quantitative and qualitative course-end assessment data indicate that students who have gone through the rubrics are receptive to the ideas, and willing to continue to learn about, apply, and be influenced by them. The reliability/validity is strongly supported by open, written student comments.

The rubrics presented in the paper are a sequential series of models (some computer based, some qualitative) or graphic presentations that are designed to represent a specific concept in support of specific learning outcomes. There are 12 models leading to 19 learning outcomes. We do not use all the models in all the classes, but pick and choose what works best for the time available and the goals of the class.

Each Model begins with a Description of the model, why we use it, why it is important, how it relates to other areas of knowledge. This is followed by a Presentation; how we develop the model in class. Finally are listed the Learning Outcomes for each model.

A pdf table of the rubric sequence is available. Below we list the steps in the learning progression in the order we present them along with the learning outcomes; each of these link to a page where the Description, Representation, and Learning Outcomes are explained, with links to any computer models or figures used during teaching.

- 1. Computational viewpoint: the idea that in a dynamic system the only way to know the outcome of an algorithm is to actually calculate it; there is no shorter route to knowing its behavior. This is not true at values below about 3.0, but becomes true at higher values. That this is not intuitively obvious is clear to the students because at higher ‘r’ values they are unable to predict the changing behavior of the system based on past behavior.

- 2. Positive/negative feedback: one central feature of all complex systems is that their behavior stems from the interplay of positive and negative feedbacks. This is a concept that students, when thrown into a natural complex system with many feedbacks operating, may have trouble grasping. The logistic system being so simple makes the influence of positive and negative feedback transparent.

- 3. ‘r’ values: ‘r’ can be translated as rate of growth, although we use ‘r’ from this point on to talk about whether the ‘r’ value of any system is high (dissipating lots of energy and/or information), or low (settling toward an equilibrium state).

- 4. Deterministic does not equal predictable: this is a very deep concept, especially if we explore its philosophical or theological roots (which in some classes we do). But, in classical science deterministic equals predictable, and predictable means it is deterministic. The logistic system is undeniably deterministic, but at higher ‘r’ values all semblance of predictability breaks down. For understanding complex systems this is the most important concept gained from exploration of the logistic system.

- 5. Bifurcations are a change in the behavior of the system and the entire range of behaviors can be summarized in one diagram. Eric Newman, at his MyPhysicsLab web site has an applet of a chaotic pendulum. In just a couple of minutes we can demonstrate that bifurcations to complexity occur in a system that, going back to Galileo, is often taken as one of the best examples of a classical system.

- 6. Instability increases with 'r'. The harder a system is pushed, the higher the ’r’ value, the more unstable and unpredictable its behavior becomes,

- 7. Self-similarity is patterns, within patterns, within patterns, so that you see complex detail at all scales of observation, all generated by an iterative process. We show examples of river drainage systems, plant branching patterns, zoom in on the Dow Jones on the stock market index, look at Earth temperature patterns during the past ice age, sea level curves, cladograms, and examples of fractally generated landscapes.

- 8. There is no typical or average size of events or objects; they come nested inside each other, patterns within patterns, within patterns. Peripheral to these discussions, but crucial when dealing with statistics, is that normal statistical analysis cannot be used on fractally organized data sets; there is no mean or standard deviation for fractal objects.

- 9. Unlike the geometry we are usually taught, most natural objects have non-whole number dimensions.

- 10. All complex systems accelerate their rate of change—bifurcations—at the same rate. This is a universality property; if this rate of change in behavior of a system is detected, the system is a complex system.

- 11. All changes—bifurcations—in a system are preceded by increasing instability in the behavior of the system. After splitting the system settles down toward stability again.

- 12. In a complex system at high ‘r’ values a difference as little as one part in a million can result in different histories for the system.

- 13. Small—low energy—events are very common, but do very little work. Large—high energy—events are very rare, but do most of the work in a system. In a complex system at high ‘r’ values a difference as little as one part in a million can result in different histories for the system.

- 14. Complex systems have behaviors that may superficially appear random, but have recognizable larger scale patterns.

Link to Description, Representation, Learning Outcomes, models and diagrams.

- 15. The general evolutionary algorithm—1) differentiate, 2) select, 3) amplify, 4) repeat—is an extremely efficient and effective method of natural selection.

- (Same as Last) 15. The general evolutionary algorithm—1) differentiate, 2) select, 3) amplify, 4) repeat—is an extremely efficient and effective method of natural selection.

- 16. Local Rules lead to Global Behavior, self organization arises spontaneously without design, or purpose, or teleological mechanisms.

- 17. All natural open systems dissipating sufficient energy evolve—self-organize—to critical, sensitive dependent states which lead to avalanches of change that follow a power law distribution.

- (Same as last.) 17. All natural open systems dissipating sufficient energy evolve—self-organize—to critical, sensitive dependent states which lead to avalanches of change that follow a power law distribution.

- 18. In a complex system everything is connected with everything else. Nothing exists in isolation from the rest, sitting in a protected niche, independent and self-sufficient.

- 19. In a complex system no one can be completely safe, with complete control over their fate. Everyone has the potential to be an innocent victim since there is no way one can fully protect oneself from external actions.